which is an isometry for the Kobayashi metrics on the domain and the codomain. The image \(\widetilde{\phi }_{(X, q)}(\mathbb {H})\)  is a complex geodesic for the Kobayashi metric on \(\mathcal {T}_g\) . Such an embedding is called the Teichmüller disc generated by the pair \((X, q)\)  Since the marking does not play any role in what follows, we drop it from the notation. A pair \((X, q)\)  is also called a half translation surface., see [1]} and [2]} for the construction of such discs. Let \(\operatorname{Stab}(\widetilde{\phi }_{(X, q)})\)  be the subgroup of \(\Gamma _g\)  which preserves the image \(\widetilde{\phi }_{(X, q)}(\mathbb {H})\)  set wise. The embedding (REF ), upon passing to the quotient by the action of \(\Gamma _g\) , induces the complex geodesic
\(f \colon \mathbb {H}\rightarrow \mathcal {T}_g \rightarrow \mathcal {M}_g\) 
